I don't see that Texas would be an independent country. If it were then extreme nationalists in the USA would be up in arms on the principle of "Once in the Union, forever in the Union". That would apply to any territory given statehood; therefore the Sandwich Islands can be given up along with Cuba and the states the CSA purchased from Mexico (none of which are "real" states.) The rest however, "George Washington will be rolling in his grave if this Benedict Arnold government gives up a single square inch of our sacred soil."
Texas is independent in the same way Poland was independent from the Soviet Union; in name only, with perhaps a few bones thrown to the administrators. US troops still occupy the important parts of Texas, US troops still have absolute authority over Texans (such as with Morrell's Equality pamphlet), and Texas can't even sneeze without the US's permission. In return, the US returns Texan POWs quicker and doesn't take hostages like in the rest of the old CSA.
Texas, in other words, is holding onto independence by the skin of its teeth... for the moment.
I agree with many people here: The CS should have gotten Arizona/New mexico during the war of succesion, and Maryland too, since the whole point of the Antietam campaign was to get Maryland to succeed. In OTL they didn't, because of the Southern defeat, but with such a wildly decisive Southern victory, they would have been annexed, and probably West virginia too, right from the beginning.
Two real reasons why not. One, the CSA didn't have any control over anything in Arizona/New Mexico (which aren't even divided), so it's hard for them to claim the land. Second, Britain was hoping to chop down the US without making it overly hostile. Since D.C. was still American, and Lee's invasion of Maryland
didn't get the support of the Marylanders that the CSA had been hoping for, it wouldn't have really been in the cards to isolate the US capital by seizing all the unwilling land around it.